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From Classical to Quantum Channels: Achieving Positive Covert Rates

Hassan ZivariFard, Xiaodong Wang, Alexei Ashikhmin

arXiv:2609.36162v1quant-phcs.ITmath.IT

TL;DR

Covert communication over memoryless channels is often limited by the square-root law, and systematic conditions for positive rates are needed. The paper uses channel geometry to characterize covertness across classical, classical-quantum, and quantum settings. It identifies channel structures under which positive covert rates are achievable.

  • Problem

    Covert transmission over classical, classical-quantum, and quantum channels generally follows the square-root law, limiting throughput, while systematic positive-rate conditions remain sparse.

    Source: §I · p. 1, A. Our Contributions · p. 2

  • Method

    The paper characterizes covertness geometrically by testing whether the innocent output distribution or state lies in the convex hull or affine span induced by communication inputs.

    Source: A. Our Contributions · pp. 2–3

  • Results

    Positive covert rates are achievable for certain classical and classical-quantum channels with sufficient input-simplex dimension, while mixed innocent states always permit a non-trivial covert ensemble in the fully quantum setting.

    Source: A. Our Contributions · p. 3, §VI · p. 38

  • Takeaways & Limitations

    Richer classical input alphabets, additional MAC transmitters, and mixed quantum innocent states can provide conditions for satisfying covertness with non-trivial inputs.

    Source: A. Our Contributions · p. 3, §VI · p. 38

Abstract

from arXiv · show

In this paper, we study the conditions under which covert communication at positive rates is feasible over discrete memoryless classical, classical-quantum, and quantum channels. For classical point-to-point channels, we show that if the dimension of the channel input probability simplex exceeds that of the channel output probability simplex, equivalently, if the input alphabet has larger cardinality than the output alphabet, then positive covert rates are achievable for certain classes of Discrete Memoryless Channels (DMCs). We further show that allowing the innocent symbol (i.e., the symbol transmitted in the no-communication mode) to be chosen appropriately can enlarge the class of DMCs for which positive covert rates are achievable. We also study covert communication over classical Multiple-Access Channels (MACs), where the additional transmitter effectively enlarges the set of available channel input pairs, and we show that the conditions required to satisfy the covertness constraint are less restrictive for MACs than for point-to-point channels. We extend these results to classical-quantum DMCs by showing that if the dimension of the input probability simplex exceeds the affine dimension of the set of output states that can be induced at the channel output, then positive covert rates are achievable for certain classes of classical-quantum DMCs. Finally, for quantum channels, we show that if the innocent state (i.e., the state transmitted in the no-communication mode) is mixed, then the covertness constraint can always be satisfied by a non-trivial input ensemble. Consequently, positive covert rates are achievable whenever the legitimate receiver can distinguish at least two states in a suitable such ensemble.

I. INTRODUCTION

Classical covert communication is usually limited by the square-root law, while positive rates have been achieved in special settings. This paper develops a geometric account of when channel structure permits positive covert rates across classical, classical-quantum, and quantum channels.

I. INTRODUCTION

  • The square-root law limits covert transmission over classical, classical-quantum, and quantum channels to order the square root of channel uses, motivating positive-rate settings.Prior mechanisms include warden uncertainty, friendly jamming, and channel-state information; the paper seeks systematic conditions for positive rates without such auxiliary resources.Source: §I · pp. 1–2

A. Our Contributions

  • For classical DMCs, geometric conditions link positive covert rates to input-alphabet richness, suitable innocent-symbol choice, and the available channel input pairs in MACs.An appropriate innocent symbol always permits a non-trivial covert input distribution for binary-output DMCs with more than two input symbols; joint MAC inputs can create opportunities unavailable under independent inputs.Source: A. Our Contributions · p. 3
  • For classical-quantum DMCs, positive covert rates are achievable for certain channels when the input probability simplex exceeds the affine dimension of the induced output states.The covertness condition matches the corresponding classical condition.Source: A. Our Contributions · p. 3
  • For fully quantum channels, a mixed innocent input state always permits a non-trivial input ensemble satisfying covertness.Positive covert rates follow when the legitimate receiver can distinguish at least two states in a suitable ensemble; the paper also studies known channels with pure innocent states.Source: A. Our Contributions · p. 3
  • The achievability scheme uses random coding, pinching, and channel resolvability, and stochastic encoding can enlarge covert capacity relative to deterministic encoding.The introduction situates these results against prior work on square-root-law limits for point-to-point DMCs and MACs, as well as special positive-rate settings.Source: A. Our Contributions · p. 3, B. State of the Art · p. 4

B. Classical-Quantum and Classical Channels

The section formulates covert communication over classical and classical-quantum channels, then characterizes when classical point-to-point channels can support positive covert rates. For binary-output channels, the condition becomes more permissive as the input alphabet grows, and choosing the innocent symbol can guarantee covertness when there are at least three inputs.

B. Classical-Quantum and Classical Channels

  • The classical-quantum setup replaces the innocent input state with an innocent input symbol while retaining the same covertness constraint.The channel output for each input symbol is represented by a density operator.Source: B. Classical-Quantum and Classical… · p. 6

III. CLASSICAL POINT-TO-POINT CHANNELS

  • The classical DMC covert-capacity theorem uses stochastic encoding with channel prefixing, which can achieve at least the deterministic-encoding covert capacity.An auxiliary random variable can provide additional design freedom by reducing I(X; Z) − I(X; Y).Source: §III · p. 7

X, WY Z|X, Y, Z

  • Positive covert rates are achievable for certain classical DMCs when the input alphabet is larger than the output alphabet.Increasing the input alphabet relative to the output alphabet progressively enlarges the class of DMCs for which covertness can be satisfied.Source: X, WY Z|X, Y, Z · p. 7

A. Discrete-Input and Binary-Output Warden Channels

  • For a binary-output DMC, covertness holds exactly when the innocent symbol’s parameter lies between the minimum and maximum parameters of the other inputs.When there are at least three inputs and the innocent symbol is freely chosen, selecting a median parameter always satisfies the condition.Source: A. Discrete-Input and Binary-Output… · p. 9
  • Increasing the input alphabet can only enlarge the set of warden output distributions and relax the covertness constraint.For continuous inputs and binary outputs, the constraint can always be satisfied.Source: A. Discrete-Input and Binary-Output… · p. 10

1) Binary-Input, Binary-Output Warden DMCs:

For binary-output warden channels, enlarging the input alphabet expands the set of output distributions available during communication, making the covertness condition less restrictive. With continuous input and a threshold detector, a non-degenerate input distribution can satisfy covertness for every positive power constraint.

1) Binary-Input, Binary-Output Warden DMCs:

  • When both binary-input symbols induce the same warden output distribution, communication is covert for any input distribution and covert capacity equals the legitimate channel’s Shannon capacity.With the innocent symbol fixed to 0, the covertness condition requires the two output parameters to be equal.Source: 1) Binary-Input, Binary-Output Warden… · p. 10

2) Ternary-Input, Binary-Output Warden DMCs:

  • The ternary-input output set contains the binary-input set, so the innocent output may lie in its interior and permit positive covert rates.The ternary covertness condition is less restrictive than the binary-input condition.Source: 2) Ternary-Input, Binary-Output Warden… · p. 11

3) Quaternary-Input, Binary-Output Warden DMCs:

  • The quaternary-input output set contains the ternary-input set, further relaxing covertness and potentially enabling positive covert rates.For fixed parameters of the first three inputs, its covertness condition is less restrictive than the binary- and ternary-input conditions.Source: 3) Quaternary-Input, Binary-Output… · p. 12

B. Continuous-Input Binary-Output Warden Channels

  • For any positive power constraint, a non-degenerate continuous-input distribution can reproduce the innocent symbol’s warden output distribution.For the threshold-detector setting, a positive covert rate is achievable for every positive average-power constraint.Source: B. Continuous-Input Binary-Output… · pp. 12–14

C. Extension to Warden DMCs with Arbitrary Output Alphabet

For arbitrary finite warden output alphabets, covertness feasibility is governed by the convex hull of output distributions induced by non-innocent inputs. If that hull is the whole output simplex, covertness is feasible for any innocent symbol, but positive rates also require receiver distinguishability.

C. Extension to Warden DMCs with Arbitrary Output Alphabet

  • Satisfying covertness alone does not guarantee a positive rate; the receiver must distinguish at least two channel inputs.If every input induces the same receiver output distribution as the innocent input, the receiver’s mutual information is zero.Source: C. Extension to Warden DMCs with… · pp. 15–16
  • When non-innocent inputs induce the entire output probability simplex, the covertness constraint can be satisfied for any innocent symbol.This is a sufficient condition; a positive covert rate additionally requires the legitimate receiver to distinguish at least two inputs.Source: C. Extension to Warden DMCs with… · pp. 15–16
  • Adding input symbols expands the convex hull of warden output distributions and may bring the innocent output distribution into that hull, enabling covertness.The section illustrates this geometry for ternary outputs, including a continuous-input channel whose induced distributions form a curve in the simplex.Source: C. Extension to Warden DMCs with… · pp. 15–16

1) Discrete-Input Ternary-Output Warden Channels:

For ternary-output warden channels, covertness feasibility can be visualized through the convex hull of induced output distributions. Enlarging the input set or choosing the innocent symbol strategically can make the innocent output reproducible; continuous-input examples show threshold conditions that suffice but are not necessary.

1) Discrete-Input Ternary-Output Warden Channels:

  • With |Z| ≥ 3, |X| > |Z| alone does not guarantee covertness for any innocent symbol, though choosing the innocent symbol appropriately can help.If a symbol’s output lies in the convex hull of the other induced outputs, choosing it as innocent makes the covertness constraint satisfiable.Source: 1) Discrete-Input Ternary-Output… · p. 17

2) Quaternary- and Quinary-Input, Ternary-Output Warden DMCs:

3) Continuous-Input Ternary-Output Warden Channels:

  • When τ2 − τ1 ≥ 2, all three ternary-simplex corners are inducible; when τ2 − τ1 > 2, each has infinitely many inducing symbols, supporting covertness for any innocent symbol.The gap condition is sufficient but not necessary: with τ1 = −0.5 and τ2 = 0.5, the gap is 1 and covertness is still feasible although not all three corners can be induced.Source: 3) Continuous-Input Ternary-Output… · p. 19

IV. EXTENSION TO MACS

A MAC enlarges the effective input alphabet through joint input pairs, creating additional ways to reproduce the warden’s no-communication output distribution. For binary-input MACs with independent inputs, the paper characterizes when a non-trivial product input can meet covertness and explains why the condition can be less restrictive than for point-to-point channels.

A. MACs with Independent Channel Inputs

  • A non-trivial independent input satisfies covertness if and only if d10 = 0, d01 = 0, or the three values d10, d01, and d11 straddle zero.Non-trivial means at least one transmitter has a non-degenerate input distribution.Source: A. MACs with Independent Channel Inputs · pp. 22–23
  • Choosing the innocent input pair freely always permits a non-trivial joint input distribution satisfying covertness for the binary-input MAC.The argument selects an innocent pair among the four input pairs and uses another pair that induces the same warden output distribution or meets the stated condition.Source: A. MACs with Independent Channel Inputs · p. 24
  • Independent inputs retain the same interval of reachable warden output distributions as the convex hull, but restrict which input distributions realize each output.Covertness also requires a reachable realization with at least one transmitter using a non-degenerate input distribution.Source: A. MACs with Independent Channel Inputs · p. 24
  • Additional joint input symbols can place warden output distributions on opposite sides of the innocent output, enabling cancellation even when the corresponding point-to-point condition fails.The MAC condition can therefore be satisfied in cases where the binary-input, binary-output point-to-point condition cannot.Source: A. MACs with Independent Channel Inputs · p. 24
  • For the example MAC, α1 = 0.5 and α2 = 0.075 satisfy covertness, whereas either transmitter alone cannot achieve a positive covert rate.The example has d10 = −0.03, d01 = 0.32, and d11 = 0.05, which straddle zero.Source: A. MACs with Independent Channel Inputs · p. 24

1) Examples:

The examples examine independent inputs when one or both warden-facing components are binary symmetric channels. A single BSC component permits a non-trivial covertness-satisfying input in the stated non-degenerate case, whereas two non-degenerate BSC components do not.

1) Examples:

  • With one BSC component, whenever d10 ≠ 0, the two relevant terms have opposite signs, guaranteeing a non-trivial independent input satisfying covertness.For the example, α1 = 1 and α2 = 1/2 let Transmitter 2 use a non-degenerate input while Transmitter 1 is fixed at its non-innocent symbol.Source: 1) Examples · p. 25
  • If d10 = 0 and d01 ≠ 0 in the one-BSC case, choosing α1 in (0, 1) and α2 = 0 also satisfies covertness.This is a degenerate-parameter case handled separately from the opposite-sign construction.Source: 1) Examples · p. 25
  • When both component channels are non-degenerate BSCs, no non-trivial independent input distribution can satisfy covertness.The only simultaneous zeros are the all-innocent input and a deterministic non-innocent input, neither of which provides a non-degenerate communication input.Source: 1) Examples · p. 25

B. MACs with Degraded Message Sets

For degraded-message MACs, allowing correlated inputs makes the covertness condition less restrictive than independent inputs, and choosing the innocent pair freely guarantees a non-trivial covert input. The section extends geometric covertness conditions to classical-quantum channels, where output states replace output distributions.

B. MACs with Degraded Message Sets

  • When the innocent input pair can be chosen freely, some choice always permits a non-trivial joint input distribution satisfying covertness.The proof chooses a pair with the second-smallest warden-output probability; either it lies between the other values or another pair produces the same value.Source: B. MACs with Degraded Message Sets · pp. 26–27
  • Adding input symbols for either transmitter adds joint input pairs and warden-output distributions, further relaxing the covertness conditions.With arbitrary joint distributions, every point in the convex hull of the four conditional warden output distributions is realizable.Source: B. MACs with Degraded Message Sets · pp. 26–27
  • With correlated inputs, the innocent warden-output probability need only lie in the closed interval generated by the three non-innocent output probabilities.Independent inputs can fail even when that range condition holds; correlation permits mixing joint input pairs directly.Source: B. MACs with Degraded Message Sets · pp. 27–28
  • For two non-degenerate BSCs with a00 = a11 and a10 = a01 ≠ a00, no non-trivial independent input is covert, while mixing (0,0) and (1,1) is covert.The equal warden-output probabilities of the two joint pairs allow their mixture to preserve the innocent output.Source: B. MACs with Degraded Message Sets · p. 28

A. Classical-Quantum Channels

  • For classical-quantum channels, covertness holds when the innocent output state lies in the convex hull of the non-innocent output states.If the output-state set has lower affine dimension than the input probability simplex, some states may lie in the convex hull of the others.Source: A. Classical-Quantum Channels · p. 29
  • When m ≥ 3 in that classical-quantum example, freely choosing the innocent symbol always permits covertness with a non-degenerate input distribution.The channel's output states lie on the line segment between Ψ0 and Ψ1.Source: A. Classical-Quantum Channels · p. 30
  • For the classical-quantum channel whose outputs lie on the segment between two distinct states, the covertness condition is identical to that of its classical counterpart, whether or not the states commute.The achievable region also allows stochastic encoding and can be strictly larger than the deterministic-encoding region.Source: A. Classical-Quantum Channels · pp. 29–30

B. Classical-Quantum MACs with Independent Inputs

For classical-quantum MACs with independent inputs, the covertness conditions match those of the corresponding classical MAC. Enlarging the input alphabet expands the available output-state set, and a sufficient channel-level condition can guarantee covertness for any innocent symbol.

B. Classical-Quantum MACs with Independent Inputs

  • For the specified classical-quantum MAC structure, independent inputs satisfy covertness under exactly the same conditions as in the corresponding classical MAC.The covertness equation reduces to the classical independent-input constraint.Source: B. Classical-Quantum MACs with… · pp. 31–32
  • Enlarging a classical-quantum channel's input alphabet expands the set of inducible output states and may bring the innocent state into the convex hull of non-innocent outputs.The inducible-state sets are nested when one input alphabet is contained in another, so their affine dimensions cannot decrease.Source: B. Classical-Quantum MACs with… · p. 32
  • If the convex hull of non-innocent output states contains the full quantum state set D(H), covertness can be satisfied for any innocent symbol.This channel-level sufficient condition applies to a finite output Hilbert space.Source: B. Classical-Quantum MACs with… · p. 33

C. Quantum Channels

For quantum channels, a mixed innocent input state always permits satisfaction of the covertness constraint. The paper also gives an example in which a two-state input ensemble satisfies covertness at a specified mixture.

C. Quantum Channels

  • An achievable rate region is derived for quantum channels, while many such channels, including the bosonic channel, are described as unable to satisfy the covertness constraint in the stated setting.The paper identifies the bosonic channel as a notable example associated with the square-root law.Source: C. Quantum Channels · pp. 33–34
  • A mixed innocent input state always allows the covertness constraint to be satisfied.Its decomposition as a convex combination of input states induces the innocent output as the same convex combination of their channel outputs.Source: C. Quantum Channels · p. 34

2. Therefore,

  • For the example with |U| = 2, the cited result states that the covertness constraint ρZ = σ0 is satisfied when α = 1.The resulting average input state is ρX = α |0⟩⟨0| + (1 − α) |1⟩⟨1|.Source: §2 · p. 35

2. At the legitimate receiver, the two

For pure innocent input states, covertness feasibility depends on whether a different input can reproduce the warden’s innocent output. The section gives examples of channels where this constraint can be satisfied and reports a positive-rate instance.

2. At the legitimate receiver, the two

  • For pure innocent states, an injective warden channel admits no non-trivial input satisfying the covertness constraint, whereas a non-injective channel may permit one and a positive covert rate.The section also gives a positive-rate example with R < 0.1379 and secret-key rate RK > 0.4109.Source: §2 · pp. 35–36

1) Entanglement-Breaking Channels:

  • In the entanglement-breaking channel example, the covertness constraint can be satisfied for the specified pure innocent state and a non-innocent input.The example specifies a pure innocent state and constructs another input state before applying Lemma 6.Source: 1) Entanglement-Breaking Channels · p. 37

2) Partial Trace Channels:

  • For the partial-trace channel, changing subsystem B while keeping subsystem A in |0⟩ leaves the warden’s output equal to the innocent output.Thus, a distinct input state satisfies the covertness constraint by Lemma 6.Source: 2) Partial Trace Channels · p. 37

3) Leakage-Reset Channel:

  • In the leakage-reset channel, both the innocent input |0⟩ and input |2⟩ induce the warden output |0⟩, satisfying the covertness constraint.The passage applies Lemma 6 to conclude feasibility.Source: 3) Leakage-Reset Channel · p. 37

VI. CONCLUSIONS

  • The conclusions extend the geometric analysis to classical-quantum channels and state that a mixed innocent state permits a non-trivial covert input for fully quantum channels.The section also reports that mixedness is essential in one example: with the specified pure innocent state, no non-trivial input ensemble can satisfy covertness.Source: §2 · p. 35, §VI · pp. 37–38

APPENDIX A

The appendices describe a random-code construction and its reliability and covertness conditions. They also outline the converse proof and a continuity argument.

A. Codebook Generation 

B. Encoding and Decoding

C. Covertness Analysis

APPENDIX B

  • The Appendix B argument states that the relevant bound vanishes as n grows and α approaches zero when R > I(U; Z).The passage attributes this step to the i.i.d. codebook construction.Source: Appendix B · p. 40

APPENDIX C

Appendix C gives a converse argument for stochastic codes, defining an expanded Qϵ rate region and bounding achievable rates. It uses single-letterization and total-variation continuity in the proof.

APPENDIX C

  • The converse considers stochastic codes for a DMC and the reliability constraint.The supplied passage identifies the codes and begins the reliability condition but does not state its full form.Source: Appendix C · p. 40
  • The proof defines an expanded Qϵ rate region and bounds any achievable rate by placing it in Qδ.The bound is stated for arbitrary positive γn and δn.Source: Appendix C · p. 41
  • The converse reduces block quantities using a uniformly random time index T and invokes total-variation monotonicity; the continuity proof for Qϵ at zero is omitted.The rate expressions compare mutual information terms involving U, Y, and Z.Source: Appendix C · pp. 42–43
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